Questions tagged [jordan-normal-form]

This tag is for questions relating to the Jordan normal form, also known as a Jordan canonical form or JCF of a matrix which is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

The Jordan normal form of a matrix is a canonical form given by a similarity transformation. The Jordan normal form consists of diagonal blocks corresponding to its generalized eigenvectors, with blocks that are themselves constant diagonal when an eigenvalue has a basis of eigenvectors and otherwise blocks that are constant diagonal + nilpotent. For more information, see this Wikipedia article.

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$3 \times 3$-matrices with the same characteristic polynomials and minimal polynomials that are not similar

I think of a counterexample to show that $3 \times 3$-matrices $A$ and $B$ can have the same characteristic polynomials and minimal polynomials without being similar: characteristic polynomials are both $(t-2)^3$. one minimal polynomial is $(t-2)^2…
user522841
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Are they all similar to each other?

I have a confusion. I have read one statement."It can be shown that the Jordan normal form of a given matrix A is unique up to the order of the Jordan blocks". But I could not understand. I have taken an exercise. If this is solved I think my…
user450210
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Let $I \neq A ∈ M_{n×n}(\mathbb{R})$ be an involutory matrix. Show that the Jordan canonical form of $A$ is a diagonal matrix.

Let $I \neq A ∈ M_{n×n}(\mathbb{R})$ be an involutory matrix. Show that the Jordan canonical form of $A$ is a diagonal matrix. I'm not sure how to do this, any solutions/hints are greatly appreciated.
1233dfv
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Find the characteristic & minimal polynomials, eigenvectors, and dimension of the eigenspace for this 6x6 jordan matrix?

3 1 0 0 0 0 0 3 1 0 0 0 0 0 3 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 I think the characteristic polynomial is: ((x-3)^3)((x-1)^3) Found by taking the number of each eigenvalue along the diagonal, 3 3's, 3 1's I think the minimal polynomial…
Bob
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Find all possible jordan normal forms of operator A

Find all possible jordan normal forms of operator A: $\mathbb{C}^7$ $\rightarrow $ $\mathbb{C}^7$ with following infos: rank$(A^3-A)^2 = 1$, $det(A-id)=12$,$rankA = 5$
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Find all possible Jordan forms of a complex matrix with parameters

Q: Given a matrix $$ A = \begin{pmatrix} a & b+c \\ b-c & -a \end{pmatrix}, $$ where $a,b,c \in \mathbb{C}$. Find all possible Jordan forms.
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Matrix and eigenvalue matrix?

If we have the eigenvalues of a 6*6 matrix with values such as = [1 1 1 2 2 3], how to write different matrix based on the same eigenvalue matrix. I really need you all to answer this question. THANK YOU.
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Find the Jordan canonical form

$N$ is a nilpotent $15\times15$ matrix over $\mathbb{R}$ such that $$\dim(\ker N) = 5, \quad \dim (\ker{N^2}) =8, \quad \dim(\ker{N^3})= 11,$$ $$\dim (\ker{N^4}) = 13, \quad \dim(\ker{N^5}) =15$$ Find the Jordan form.
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Inverse Jordan Decomposition Matlab code?

I did Jordan decomposition of a matrix by using this code: A = [1 -3 -2; -1 1 -1; 2 4 5]; [V, J] = jordan(A) Now I need to do inverse Jordan decomposition to get original matrix A. How can I do that in Matlab?
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$T:f(x)\to f(x-1)+x^3f'''(x)/3$ Find the Jordan normal form and a Jordan basis for $T$.

Let $T\in \mathcal{L}(\mathcal{P_3}(\mathbb{C})$ be the operator $$T:f(x)\to f(x-1)+\frac{x^3f'''(x)}{3}$$ Find the Jordan normal form and a Jordan basis for $T$.
kile
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Jordan matrix two looks

i have question. I have something like this: $\begin{bmatrix} -2 & 2 \\ 1 & 3 \\ \end{bmatrix}$ $\lambda_{1} = -1$ $\lambda_{2} = -4$ When jordan matrix looks like this: $\begin{bmatrix} -1 & 0 \\ 0 & -4\\ \end{bmatrix}$ And when to…
Hadson
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Is every Jordan block diagonalisable?

Is every Jordan block diagonalisable? I need to also give a short justification. Can anyone lend a hand?
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